Nonparametric pricing and hedging of exotic derivatives

TL;DR

This paper introduces signature payoffs, a new class of derivatives that enable nonparametric pricing and hedging of exotic derivatives using market data, with demonstrated accuracy and computational efficiency.

Contribution

It proposes a novel nonparametric approach using signature payoffs to price and hedge exotic derivatives based on existing market prices.

Findings

Accurate pricing and hedging achieved on real and simulated data.

Method is computationally tractable and scalable.

Signature payoffs effectively approximate exotic derivatives.

Abstract

In the spirit of Arrow-Debreu, we introduce a family of financial derivatives that act as primitive securities in that exotic derivatives can be approximated by their linear combinations. We call these financial derivatives signature payoffs. We show that signature payoffs can be used to nonparametrically price and hedge exotic derivatives in the scenario where one has access to price data for other exotic payoffs. The methodology leads to a computationally tractable and accurate algorithm for pricing and hedging using market prices of a basket of exotic derivatives that has been tested on real and simulated market prices, obtaining good results.

Figures

Equations

$$T(({\mathbb{R}}^d)):=\{\mathbf{a}=(a_0,a_1,\dots ,a_n,\dots )|a_n\in ({\mathbb{R}}^d{)}^{\otimes n}\}.$$

$$T^{(N)}({\mathbb{R}}^d):=\{\mathbf{a}=(a_n{)}_{n=0}^{\infty }|a_n\in ({\mathbb{R}}^d{)}^{\otimes n}\text{mbox}and{a}_n=0\forall n\ge N\}\subset T(({\mathbb{R}}^d)),$$

$$T({\mathbb{R}}^d):=\bigcup _{n\ge 0}{T}^{(n)}({\mathbb{R}}^d)\subset T(({\mathbb{R}}^d)).$$

$$\mathbf{a}+\mathbf{b}:=(a_i+b_i{)}_{i=0}^{\infty },$$

$$\mathbf{a}\otimes \mathbf{b}:={\left(\sum _{k=0}^i{a}_k\otimes b_{i-k}\right)}_{i=0}^{\infty }.$$

$$\{e_{i_1}\otimes \dots \otimes e_{i_n}\mid i_j\in \{1,\dots ,d\}\text{mbox}forj=1,\dots ,n\}$$

$$\{e_{i_1}^{\ast }\otimes \dots \otimes e_{i_n}^{\ast }\mid i_j\in \{1,\dots ,d\}\text{mbox}forj=1,\dots ,n\}.$$

$$e_{i_{1}}^{\ast}\otimes\ldots e_{i_{n}}^{\ast}\in T((\mathbb{R}^{d})^{\ast})\longleftrightarrow{\color[rgb]{0,0,1}\mathbf{i_{1}}}\ldots{\color[rgb]{0,0,1}\mathbf{i_{n}}}\in\mathcal{W}(\mathcal{A}_{d})$$

$${\color[rgb]{0,0,1}\mathbf{wv}}:={\color[rgb]{0,0,1}\mathbf{i_{1}}}\ldots{\color[rgb]{0,0,1}\mathbf{i_{n}j_{1}}}\ldots{\color[rgb]{0,0,1}\mathbf{j_{k}}}\in\mathcal{W}(\mathcal{A}_{d}).$$

$${\color[rgb]{0,0,1}\mathbf{ua}}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup{\color[rgb]{0,0,1}\mathbf{vb}}=({\color[rgb]{0,0,1}\mathbf{u}}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup{\color[rgb]{0,0,1}\mathbf{vb}}){\color[rgb]{0,0,1}\mathbf{a}}+({\color[rgb]{0,0,1}\mathbf{ua}}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup{\color[rgb]{0,0,1}\mathbf{v}}){\color[rgb]{0,0,1}\mathbf{b}},$$

$${\color[rgb]{0,0,1}\mathbf{w}}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup{\color[rgb]{0,0,1}\mathbf{\varnothing}}={\color[rgb]{0,0,1}\mathbf{\varnothing}}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup{\color[rgb]{0,0,1}\mathbf{w}}={\color[rgb]{0,0,1}\mathbf{w}}$$

$$P^{\bigsqcup \bigsqcup }(\ell ):=a_0+a_1\bigsqcup \bigsqcup +a_2\bigsqcup {\bigsqcup }^{\bigsqcup \bigsqcup 2}+\dots +a_n\ell {\ell }^{\bigsqcup \bigsqcup n}\forall \ell \in T(({\mathbb{R}}^d{)}^{\ast })$$

$${\mathbb{Z}}^{<\infty }:[0,T{]}^2$$

$$(s,t)$$

$${\mathbb{Z}}_{s,t}^n:={\int }_{s<u_1<\dots <u_k<t}d{Z}_{u_1}\otimes \dots \otimes d{Z}_{u_k}\in ({\mathbb{R}}^d{)}^{\otimes n}.$$

$${\mathbb{Z}}^{\le N}:[0,T{]}^2$$

$$(s,t)$$

$${\mathbb{Z}}^{<\infty }:[0,T{]}^2$$

$$(s,t)$$

$${\mathbb{Z}}_{s,t}^n:={\int }_{0<u_1<\dots <u_k<T}\circ d{Z}_{u_1}\otimes \dots \otimes \circ d{Z}_{u_k}\in ({\mathbb{R}}^d{)}^{\otimes n},$$

$${\mathbb{Z}}^{\le N}:[0,T{]}^2$$

$$(s,t)$$

$$⟨{\ell }_1,{\mathbb{Z}}^{<\infty }⟩⟨{\ell }_2,{\mathbb{Z}}^{<\infty }⟩=⟨{\ell }_1\bigsqcup \bigsqcup {\ell }_2,{\mathbb{Z}}^{<\infty }⟩.$$

$${\mathbb{Z}}^{LL,\le 2}:=\left(1,({\mathbb{Z}}^1,{\mathbb{Z}}^1),\left(\begin{array}{cc}{\mathbb{Z}}^2& {\mathbb{Z}}^2-\frac{1}{2}⟨Z⟩\\ {\mathbb{Z}}^2+\frac{1}{2}⟨Z⟩& {\mathbb{Z}}^2\end{array}\right)\right)\in G\Omega ([0,T];{\mathbb{R}}^{2d})$$

$${\hat{\Omega }}_T:={\overline{\{{\hat{\mathbb{X}}}^{<\infty }:X:[0,T]\to \mathbb{R}\text{mbox}issmoothand{X}_0=1\}}}^{d_{p-var}}\subset T(({\mathbb{R}}^2)),$$

$${\hat{\Omega }}_T^{LL}:=\{{\hat{\mathbb{X}}}^{LL,<\infty }:({\hat{\mathbb{X}}}^{\le 2},⟨\hat{X}⟩)\text{mbox}isalead-lagpathand{X}_0=1\}\subset T(({\mathbb{R}}^4)),$$

$$F({\hat{\mathbb{X}}}^{LL,<\infty })=⟨f,{\hat{\mathbb{X}}}_{0,T}^{LL,<\infty }⟩.$$

$${\mathcal{T}}^q({\Lambda }_T):=\left\{\theta \in \mathcal{T}({\Lambda }_T):\mathbb{E}\left[{\left|{\int }_0^T\theta (\hat{\mathbb{X}}{|}_{[0,t]}^{<\infty })d{X}_t\right|}^q\right]<\infty \right\}.$$

$${\mathcal{T}}_{sig}({\Lambda }_T):=\{\theta \in \mathcal{T}({\Lambda }_T)|\exists \ell \in T(({\mathbb{R}}^2{)}^{\ast })\text{mbox}suchthat\theta (\hat{\mathbb{X}}{|}_{[0,t]}^{<\infty })=⟨\ell ,{\hat{\mathbb{X}}}_{0,t}^{<\infty }⟩\forall \hat{\mathbb{X}}{|}_{[0,t]}^{<\infty }\in {\Lambda }_T\}.$$

$$\int_{0}^{T}\langle\ell,\widehat{\mathbb{X}}_{0,t}^{<\infty}\rangle dX_{t}=\langle\ell{\color[rgb]{0,0,1}\mathbf{4}},\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle,$$

$$\underset{\theta \in {\mathcal{T}}^q({\Lambda }_T)}{\inf }\mathbb{E}\left[P\left(F({\hat{\mathbb{X}}}^{LL,<\infty })-p_0-{\int }_0^T\theta (\hat{\mathbb{X}}{|}_{[0,t]}^{<\infty })d{X}_t\right)\right].$$

$$\underset{\ell \in T(({\mathbb{R}}^2{)}^{\ast })}{\inf }\mathbb{E}\left[P\left(⟨f,{\hat{\mathbb{X}}}_{0,T}^{LL,<\infty }⟩-p_0-{\int }_0^T⟨\ell ,{\hat{\mathbb{X}}}_{0,t}^{<\infty }⟩d{X}_t\right)\right].$$

$$\inf_{\ell\in T((\mathbb{R}^{2})^{\ast})}\left\langle P^{\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup}(f-p_{0}{\color[rgb]{0,0,1}\mathbf{\varnothing}}-\ell{\color[rgb]{0,0,1}\mathbf{4}}),\mathbb{E}\left[\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\right]\right\rangle.$$

Full text

1 Introduction

Arrow-Debreu securities [Arr73, Deb87] are idealised and basic securities that pay one unit of numeraire for a particular market state at a specific future time, and pay nothing otherwise. These securities are primitive in the sense that the cash flow of any derivative can be approximately written in terms of Arrow-Debreu securities. In this paper, we identify a family of primitive securities for path-dependent exotic derivatives. Analogously to Arrow-Debreu securities, cash flows of exotic derivatives can be approximated by linear combinations of these primitive securities.

When one has only limited access to market prices for a class of financial products, one may be interested in studying if knowledge of these prices can be leveraged to price other financial products in an arbitrage-free market [ASL98, DFW98, HLP94]. For example, if one is able to price zero-coupon bonds, one could deduce the price of other coupon-bearing bonds by writing the cash flows as combinations of zero-coupon bonds. Similarly, if prices of European call and put options are observable in the market, it was shown in [BL78] that this information is enough to price any European contingent claim, by writing such contingent claims in terms of put and call options.

In this paper, we take this idea a step further by showing that knowledge of prices of enough exotic derivatives is sufficient to accurately derive prices and hedging strategies of other exotic derivatives in a nonparametric manner. First, in the spirit of Arrow-Debreu [Arr73, Deb87], we approximate exotic derivatives in terms of simpler payoffs called signature payoffs (Definition 3.6). Then, we infer a certain quantity from the market: the implied expected signature (introduced in Section 6.1). This procedure, which is model-free in nature, is empirically demonstrated in Section 6.

Signature payoffs are a family of path-dependent derivatives defined in terms of certain iterated integrals and, because of this, signature derivatives contain a lot of information about all possible dynamic trading strategies.

When one buys or sells a financial derivative on one or several assets, one is immediately exposed to certain risks. If this risk is unwanted, one may be interested in offsetting it by trading the underlying assets. This problem, known as hedging, is a classical problem in mathematical finance ([BS73, Mer73, Foe85]).

In idealised markets that are complete and frictionless, by definition it is possible to perfectly hedge such financial derivatives or payoffs. Therefore, in these cases the risk can theoretically be completely eliminated by following the hedging strategy.

However, transaction costs and other market frictions make real markets incomplete and hence it is not possible, in general, to perfectly hedge any given payoff. On top of that, market frictions such as transaction costs or liquidity constraints reduce the trader’s ability to hedge. In these cases, one can try to find a hedging strategy that is optimal in the sense that it minimises a certain cost function. This cost function would be chosen by the trader, depending on her risk preferences.

An example of such optimisation problems is the mean-variance problem, where one wants to minimise the $L^{2}$ norm of the profits and losses (P&L) of the trading strategy ([Sch10, DR91, Sch92, DMKR95]). This risk measure penalises any differences between the payoff and the corresponding hedging strategy. In particular, this risk preference penalises profits as well as losses. If one doesn’t wish to penalise profits, the exponential utility function $x\mapsto\exp(-\lambda x)$ can be used instead, where $\lambda>0$ is the risk-tolerance parameter ([DGR*+*02, GH02]).

An open problem with obvious practical applications is how one could find a minimiser (or minimising sequence) for these optimal hedging problems. The paper [HKK06] addresses this question for the mean-variance problem for vanilla options on Lévy processes, where the authors give a semi-explicit solution. This was later extended in [GOR14], where the authors provide an algorithm for mean-variance hedging of vanilla options. In [JMSS12], on the other hand, the authors use BSDEs to try to get the optimal hedge for the mean-variance problem in the frictionless framework. Nevertheless, in general the optimal hedge seems to be difficult to find in practice. In [HLP94], on the other hand, the authors use neural networks to price and hedge European options. In [BGTW19] this was extended to exotic derivatives, where the authors try to solve the optimisation problem by approximating the optimal hedging strategy with deep learning. However, this approach will, in general, only find local minima and not the global minima. Moreover, the training process can be computationally expensive.

In this paper, we address the optimal hedging problem of minimising the expectation of a polynomial on the P&L. In particular, when the polynomial is chosen to be $x^{2}$, the classical mean-variance optimal hedging problem is retrieved. Stating the problem for general polynomials allows us to address the optimal hedging problem for the exponential utility function as well.

We make use of signatures from rough path theory ([LCL07]) to reduce the general optimal hedging problem to a finite-dimensional optimisation problem that is computationally solvable. This is done in Section 4 by solving a linearised version of the problem first (see Theorem 4.3) and showing then that it suffices to solve this linearised problem to solve the original one (see Theorem 4.7). This method is then described in Algorithm 1. We assume that the price process $X$ and volatility $\langle X\rangle$ of the underlying asset are such that $\mathbb{X}^{LL}:=((X,X),\langle X\rangle)$ is a 2-dimensional geometric rough path [LCL07] which we call the lead-lag price path. It is shown in [FHL16] that almost all sample paths $X$ of continuous semimartingales, together with the quadratic variation $\langle X\rangle$, have this geometric rough path property. However, we do not impose any model on the dynamics of the asset, so that our approach is model-free. Moreover, we show in Section 6 that our methodology can be applied from market data without making modelling assumptions.

The signature of a path is a transformation of path space that, in certain ways, behaves similarly to the Taylor expansion. Real-valued continuous functions on $\mathbb{R}^{d}$ are well approximated by linear functions on some polynomial basis. Similarly, real-valued continuous functions on some path space are well-approximated by linear functions on signatures (Lemma 4.5). This was already leveraged in the context of finance in [PA18], where payoffs were priced by writing them as linear functions on signatures. Signatures are concise and informative feature sets for paths, and as a consequence they have been used in machine learning in contexts other than finance, such as in mental health, handwriting recognition and gesture recognition ([Gra13, XSJ*+*18, LZJ17, LJY17, YLN*+*17, PASG*+*18]).

In Section 5 we solve several extensions of the original problem. For instance, in Sections 5.3 and 5.4 we solve the problem with market frictions – namely transaction costs and liquidity constraints. In Section 5.2, on the other hand, we study the semi-static hedging problem where the trader has access to a basket of derivatives for static hedging. Finally, in Section 5.5 we show how the optimal hedge can be found when the agent starts trading at a positive time after inception of the derivative.

Our approach is model-free in the sense that no model is assumed for the market dynamics. We show in Section 6 how our methodology can be applied using market data, without attempting to model the underlying asset’s price process. This is done using the implied expected signature, an object that is discussed in Section 6.1. We first use the implied expected signature to predict market prices of exotic payoffs in Section 6.2, and then to hedge these payoffs in Section 6.3.

In certain cases, however, one is interested in modelling the price path with a certain stochastic process. This setting is studied in Section 7, where we carry out some numerical experiments for a variety of payoffs and market models, obtaining good results.

2 Signatures

In this section we will introduce the rough path theory tools that will be need in this paper. A full introduction to rough path theory, however, is beyond the scope of this paper – we refer to [LCL07] for a detailed review of rough path theory.

2.1 Tensor algebra

Signatures take value on a certain graded space: the tensor algebra. We will now define this space and its algebraic structure.

Definition 2.1**.**

Let $d\geq 1$. We define the extended tensor algebra over $\mathbb{R}^{d}$ by

[TABLE]

Similarly, we define truncated tensor algebra of order $N\in\mathbb{N}$ and the tensor algebra, denoted by $T^{(N)}(\mathbb{R}^{d})$ and $T(\mathbb{R}^{d})$ respectively, by

[TABLE]

[TABLE]

Intuitively, the extended tensor algebra $T((\mathbb{R}^{d}))$ is the space of all sequences of tensors, $T(\mathbb{R}^{d})$ is the space of all finite sequences of tensors and $T^{(N)}(\mathbb{R}^{d})$ is the space of all sequences of length $N$ of tensors.

We equip $T((\mathbb{R}^{d}))$ with two operations: a sum $+$ and a product $\otimes$. These are defined, for $\mathbf{a}=(a_{i})_{i=0}^{\infty},\mathbf{b}=(b_{i})_{i=0}^{\infty}\in T((\mathbb{R}^{d}))$, by:

[TABLE]

We also define the action on $\mathbb{R}$ given by $\lambda\mathbf{a}:=(\lambda a_{i})_{i=0}^{\infty}$ for all $\lambda\in\mathbb{R}$. These operations induce analogous operations on $T(\mathbb{R}^{d})$ and $T^{N}(\mathbb{R}^{d})$.

Let $\{e_{1},\ldots,e_{d}\}\subset\mathbb{R}^{d}$ be a basis for $\mathbb{R}^{d}$, and let $\{e_{1}^{\ast},\ldots,e_{d}^{\ast}\}\subset(\mathbb{R}^{d})^{\ast}$ be the associated dual basis for the dual space $(\mathbb{R}^{d})^{\ast}$. This induces a basis for $(\mathbb{R}^{d})^{\otimes n}$:

[TABLE]

and a basis of $((\mathbb{R}^{d})^{\ast})^{\otimes n}$:

[TABLE]

Bases for $T((\mathbb{R}^{d}))$ and $T((\mathbb{R}^{d})^{\ast})$ are then canonically constructed from the bases for $(\mathbb{R}^{d})^{\otimes n}$ and $((\mathbb{R}^{d})^{\ast})^{\otimes n}$, respectively.

We will identify the dual space $T((\mathbb{R}^{d})^{\ast})$ with the space of all words. Consider the alphabet \mathcal{A}_{d}:=\{{\color[rgb]{0,0,1}\mathbf{1}},{\color[rgb]{0,0,1}\mathbf{2}},\ldots,{\color[rgb]{0,0,1}\mathbf{d}}\}, which consists of $d$ letters. We make the following identification:

[TABLE]

where $\mathcal{W}(\mathcal{A}_{d})$ is the real vector space of all words with alphabet $\mathcal{A}_{d}$. The empty word will be denoted by {\color[rgb]{0,0,1}\mathbf{\varnothing}}\in\mathcal{W}(\mathcal{A}_{d}). We then have the identification $T((\mathbb{R}^{d})^{\ast})\cong\mathcal{W}(\mathcal{A}_{d})$.

Example 2.2*.*

We will now include a few examples in $\mathbb{R}^{2}$. In this case, the alphabet is given by \mathcal{A}_{2}=\{{\color[rgb]{0,0,1}\mathbf{1}},{\color[rgb]{0,0,1}\mathbf{2}}\}.

Set $\mathbf{a}:=2+e_{1}-e_{2}\otimes e_{1}\in T((\mathbb{R}^{2}))$. Then, \langle{\color[rgb]{0,0,1}\mathbf{\varnothing}},\mathbf{a}\rangle=2. 2. 2.

Set $\mathbf{a}:=e_{1}\otimes e_{2}-e_{2}\otimes e_{1}\in T((\mathbb{R}^{2}))$. Then, \langle{\color[rgb]{0,0,1}\mathbf{12}}+{\color[rgb]{0,0,1}\mathbf{21}},\mathbf{a}\rangle=1-1=0. 3. 3.

Set $\mathbf{a}:=-1+3e_{1}^{\otimes 3}\in T((\mathbb{R}^{2}))$. Then, \langle 2\cdot{\color[rgb]{0,0,1}\mathbf{\varnothing}}+{\color[rgb]{0,0,1}\mathbf{2}}+{\color[rgb]{0,0,1}\mathbf{111}},\mathbf{a}\rangle=2\cdot(-1)+0+3=1.

Two important algebraic operations on words are the sum and concatenation. The sum of two words {\color[rgb]{0,0,1}\mathbf{w}},{\color[rgb]{0,0,1}\mathbf{v}}\in\mathcal{W}(\mathcal{A}_{d}) is just the formal sum {\color[rgb]{0,0,1}\mathbf{w}}+{\color[rgb]{0,0,1}\mathbf{v}}\in\mathcal{W}(\mathcal{A}_{d}). The concatenation of {\color[rgb]{0,0,1}\mathbf{w}}={\color[rgb]{0,0,1}\mathbf{i_{1}}}\ldots{\color[rgb]{0,0,1}\mathbf{i_{n}}},{\color[rgb]{0,0,1}\mathbf{v}}={\color[rgb]{0,0,1}\mathbf{j_{1}}}\ldots{\color[rgb]{0,0,1}\mathbf{j_{m}}}\in\mathcal{W}(\mathcal{A}_{d}) is defined by

[TABLE]

This operation is extended by bilinearity to all of $\mathcal{W}(\mathcal{A}_{d})$. With some abuse of notation, we will use concatenation on $\mathcal{W}(\mathcal{A}_{d})$ and $T((\mathbb{R}^{d})^{\ast})$ interchangeably, in the sense that we will sometimes write \ell{\color[rgb]{0,0,1}\mathbf{w}}\in T((\mathbb{R}^{d})^{\ast}) for $\ell\in T((\mathbb{R}^{d})^{\ast})$, {\color[rgb]{0,0,1}\mathbf{w}}\in\mathcal{W}(\mathcal{A}_{d}) to denote the concatenation of the element in $\mathcal{W}(\mathcal{A}_{d})$ associated to $\ell$ and the word {\color[rgb]{0,0,1}\mathbf{w}}.

Example 2.3*.*

Take the alphabet \mathcal{A}_{3}=\{{\color[rgb]{0,0,1}\mathbf{1}},{\color[rgb]{0,0,1}\mathbf{2}},{\color[rgb]{0,0,1}\mathbf{3}}\}.

Let {\color[rgb]{0,0,1}\mathbf{w}}={\color[rgb]{0,0,1}\mathbf{132}} and {\color[rgb]{0,0,1}\mathbf{v}}={\color[rgb]{0,0,1}\mathbf{133}}. Then, {\color[rgb]{0,0,1}\mathbf{wv}}={\color[rgb]{0,0,1}\mathbf{132133}}. 2. 2.

Take {\color[rgb]{0,0,1}\mathbf{w}}={\color[rgb]{0,0,1}\mathbf{312}}, {\color[rgb]{0,0,1}\mathbf{v}}={\color[rgb]{0,0,1}\mathbf{2}} and {\color[rgb]{0,0,1}\mathbf{u}}={\color[rgb]{0,0,1}\mathbf{23}}. Then, ({\color[rgb]{0,0,1}\mathbf{w}}+{\color[rgb]{0,0,1}\mathbf{v}}){\color[rgb]{0,0,1}\mathbf{u}}={\color[rgb]{0,0,1}\mathbf{31223}}+{\color[rgb]{0,0,1}\mathbf{223}}.

Another operation one can define on words, which will be key in this paper, is the shuffle product:

Definition 2.4** (Shuffle product).**

The shuffle product $\phantom{}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup\phantom{}:\mathcal{W}(\mathcal{A}_{d})\times\mathcal{W}(\mathcal{A}_{d})\to\mathcal{W}(\mathcal{A}_{d})$ is defined inductively by

[TABLE]

[TABLE]

for all words {\color[rgb]{0,0,1}\mathbf{u}},{\color[rgb]{0,0,1}\mathbf{v}} and letters {\color[rgb]{0,0,1}\mathbf{a}},{\color[rgb]{0,0,1}\mathbf{b}}\in\mathcal{A}_{d}, which is then extended by bilinearity to $\mathcal{W}(\mathcal{A}_{d})$. With some abuse of notation, the shuffle product on $T((\mathbb{R}^{d})^{\ast})$ induced by the shuffle product on words will also be denoted by $\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup\phantom{}$.

The shuffle product gets its name from riffle shuffling of cards. If one wants to shuffle two piles of cards {\color[rgb]{0,0,1}\mathbf{w}} and {\color[rgb]{0,0,1}\mathbf{v}}, then {\color[rgb]{0,0,1}\mathbf{w}}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup{\color[rgb]{0,0,1}\mathbf{v}} is the sum of all possible outcomes from riffle shuffling.

Example 2.5*.*

For the alphabet \mathcal{A}_{4}=\{{\color[rgb]{0,0,1}\mathbf{1}},{\color[rgb]{0,0,1}\mathbf{2}},{\color[rgb]{0,0,1}\mathbf{3}},{\color[rgb]{0,0,1}\mathbf{4}}\},

{\color[rgb]{0,0,1}\mathbf{12}}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup{\color[rgb]{0,0,1}\mathbf{3}}={\color[rgb]{0,0,1}\mathbf{123}}+{\color[rgb]{0,0,1}\mathbf{132}}+{\color[rgb]{0,0,1}\mathbf{312}}. 2. 2.

{\color[rgb]{0,0,1}\mathbf{12}}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup{\color[rgb]{0,0,1}\mathbf{34}}={\color[rgb]{0,0,1}\mathbf{1234}}+{\color[rgb]{0,0,1}\mathbf{1324}}+{\color[rgb]{0,0,1}\mathbf{1342}}+{\color[rgb]{0,0,1}\mathbf{3124}}+{\color[rgb]{0,0,1}\mathbf{3142}}+{\color[rgb]{0,0,1}\mathbf{3412}}.

Definition 2.6**.**

Let $P=a_{0}+a_{1}x+\ldots+a_{n}x^{n}\in\mathbb{R}[x]$ be a polynomial on one variable. Then, $P$ induces a map $P^{\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup}$ defined by

[TABLE]

where $\ell^{\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup i}:=\underbrace{\ell\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup\ldots\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup\ell}_{i}$ for each $i\in\mathbb{N}$.

2.2 Signatures

We will now define the signature of a smooth path, together with the notion of a geometric rough path, first introduced in [Lyo98]. In our framework, we will model the price path of an asset by a geometric rough path. Given that semimartingales are geometric rough paths ([Lyo98]), our framework will in particular include all semimartingales, so that for simplicity the reader may want to have that important example in mind when results for geometric rough paths are stated. Working with geometric rough paths will allow us to consider a more general, model-free framework.

Definition 2.7** (Signature).**

Let $Z:[0,T]\to\mathbb{R}^{d}$ be smooth. The signature of $Z$ is defined by

[TABLE]

where

[TABLE]

Similarly, the truncated signature of order $N\in\mathbb{N}$ is defined by

[TABLE]

If we do not specify the interval $[s,t]$ and just mention the signature of $Z$, we will implicitly be referring to $\mathbb{Z}_{0,T}^{<\infty}$.

Definition 2.8** (Geometric rough path).**

$\mathbb{Z}^{\leq 2}:[0,T]^{2}\to T^{(2)}(\mathbb{R}^{d})$* is said to be a geometric rough path ([LCL07]) if it is the limit (under the $p$-variation distance, [LCL07, Definition 1.5]) of truncated signatures of order 2 of smooth paths. The space of all geometric rough paths will be denoted by $G\Omega([0,T];\mathbb{R}^{d})$. A geometric rough path $\mathbb{Z}^{\leq 2}\in G\Omega([0,T];\mathbb{R}^{d})$ can be (uniquely) extended to $\mathbb{Z}^{<\infty}:[0,T]^{2}\to T((\mathbb{R}^{d}))$, which will be called its signature ([LCL07, Theorem 3.7]).*

Example 2.9*.*

Semimartingales are geometric rough paths almost surely. Given a continuous semimartingale $Z:[0,T]\to\mathbb{R}^{d}$, its signature is given by

[TABLE]

where

[TABLE]

with the integrals understood in the sense of Stratonovich. Similarly, the truncated signature of order $N\in\mathbb{N}$ is defined by

[TABLE]

We will now include a few examples to provide an intuition about the iterated integrals that define signatures.

Example 2.10*.*

Let $Z=(Z^{1},Z^{2})$ be a continuous semimartingale on $\mathbb{R}^{2}$. Recalling the notation of words introduced in the previous section, we have:

\langle{\color[rgb]{0,0,1}\mathbf{\varnothing}},\mathbb{Z}_{0,T}^{<\infty}\rangle=1. 2. 2.

\langle{\color[rgb]{0,0,1}\mathbf{1}},\mathbb{Z}_{0,T}^{<\infty}\rangle=\int_{0}^{T}\circ dZ_{t}^{1}=Z_{T}^{1}-Z_{T}^{1}. 3. 3.

\langle{\color[rgb]{0,0,1}\mathbf{22}},\mathbb{Z}_{0,T}^{<\infty}\rangle=\int_{0}^{T}\int_{0}^{t}\circ dZ_{s}^{2}\circ dZ_{t}^{2}=\int_{0}^{t}(Z_{t}^{2}-Z_{0}^{2})\circ dZ_{t}^{2}=\frac{1}{2}(Z_{T}-Z_{0})^{2}. 4. 4.

\langle{\color[rgb]{0,0,1}\mathbf{12}},\mathbb{Z}_{0,T}^{<\infty}\rangle=\int_{0}^{T}\int_{0}^{t}\circ dZ_{s}^{1}\circ dZ_{t}^{2}=\int_{0}^{T}(Z_{t}^{1}-Z_{0}^{1})\circ dZ_{t}^{2}. 5. 5.

Take $\ell\in T((\mathbb{R}^{2})^{\ast})$. Then, \langle\ell{\color[rgb]{0,0,1}\mathbf{1}},\mathbb{Z}_{0,T}^{<\infty}\rangle=\int_{0}^{T}\langle\ell,\mathbb{Z}_{0,t}^{<\infty}\rangle\circ dZ_{t}^{1}.

We will now state two properties of signatures that will have a crucial role in this paper. The first property, the shuffle product property, states that the product of two linear functions on the signature is a new linear function on the signature. The second property is a uniqueness result: the signature of a path is unique.

Lemma 2.11** (Shuffle product property, [LCL07]).**

Let $\mathbb{Z}^{\leq 2}\in G\Omega([0,T];\mathbb{R}^{d})$ be a geometric rough path, and let $\ell_{1},\ell_{2}\in T((\mathbb{R}^{d})^{\ast})$ be two linear functionals. Then,

[TABLE]

Lemma 2.12** (Uniqueness of signatures, [BGLY16]).**

Let $\mathbb{Z}^{\leq 2}\in G\Omega([0,T];\mathbb{R}^{d})$ be a geometric rough path. Its signature over $[0,T]$, $\mathbb{Z}_{0,T}^{<\infty}$, uniquely determines $\mathbb{Z}^{\leq 2}$, up to tree-like equivalences (see [BGLY16, Definition 1.1] for a definition).

Corollary 2.13**.**

Let $\in\mathbb{Z}^{\leq 2}\in G\Omega([0,T];\mathbb{R}^{d})$ be a geometric rough path. Assume there exists a linear function $\ell\in T^{(2)}((\mathbb{R}^{d})^{\ast})$ such that $t\mapsto\mathbb{\langle}\ell,\mathbb{Z}_{0,t}^{\leq 2}\rangle$ is strictly monotone. Then, the signature $\mathbb{Z}_{0,T}^{<\infty}$ uniquely determines $\mathbb{Z}^{\leq 2}$.

2.3 Lead-lag path

In our framework, prices are going to be given by a path $Z:[0,T]\to\mathbb{R}^{d}$, which will be assumed to be a geometric rough path, and a volatility $\langle Z\rangle:[0,T]\to\mathbb{R}^{d\times d}$. For simplicity, the reader may want to think of $Z$ being a semimartingale and $\langle Z\rangle$ the quadratic variation of $Z$, as this is included in our framework.

Definition 2.14** (Lead-lag path).**

A lead-lag path is a pair $(\mathbb{Z}^{\leq 2},\langle Z\rangle)$ with $\mathbb{Z}^{\leq 2}\in G\Omega([0,T];\mathbb{R}^{d})$ a geometric rough path (Definition 2.8) and $\langle Z\rangle:[0,T]^{2}\to\mathbb{R}^{d\times d}$ such that $\langle Z\rangle$ is symmetric and

[TABLE]

is a geometric rough path on $\mathbb{R}^{2d}$. $\mathbb{Z}^{LL,<\infty}$ will be called the signature of the lead-lag path $(\mathbb{Z}^{\leq 2},\langle Z\rangle)$.

Appendix A includes a discussion on how to obtain in practice the lead-lag path associated with a semimartingale or discrete data, as well as its signature.

Example 2.15*.*

A continuous semimartingale $Z:[0,T]\to\mathbb{R}^{d}$ induces a geometric rough path $\mathbb{Z}^{\leq 2}$, as shown in Example 2.9. This geometric rough path is given by certain Stratonovich iterated integrals. $\mathbb{Z}^{\leq 2}$, together with the quadratic variation $\langle Z\rangle_{s,t}$ of $Z$ over $[s,t]$, induces the lead-lag path $(\mathbb{Z}^{\leq 2},\langle Z\rangle)$. Such lead-lag paths were considered in [FHL16]. As we will see in Lemma 3.11, if the price process is a semimartingale certain Itô integrals against the semimartingale can be written as integrals against the lead-lag process.

3 Framework

3.1 The market

For simplicity, we will consider the case where there is only a single underlying risky asset. However, the authors would like to emphasise that all the results in this paper can be readily extended to the multi-asset case. In the sequel, we will model the (discounted) price path of the underlying asset by a continuous curve in $\mathbb{R}$, $X:[0,T]\to\mathbb{R}$. We will denote the augmentation of $X$ (as in [PA18]) by $\widehat{X}_{t}:=(t,X_{t})\in\mathbb{R}^{2}$. Without loss of generality, we will assume that the initial price of the asset is given by $X_{0}=1$. In our framework, the market will be given by the price path $\widehat{X}:[0,T]\to\mathbb{R}^{2}$, together with a volatility process $\langle\widehat{X}\rangle:[0,T]\to\mathbb{R}^{2\times 2}$. Almost all paths of a semimartingale $X:[0,T]\to\mathbb{R}$ are included in this framework, in which case the volatility process $\langle\widehat{X}\rangle_{t}$ is just the quadratic variation of $\widehat{X}$. Tick-data is also included in this framework, as it induces a lead-lag path ([FHL16]). However, our approach is model-free in the sense that we do not impose any model on the price path, nor do we assume it is a realisation of a semimartingale. Nevertheless, for simplicity the reader may think of $X$ as a semimartingale. We will now introduce the precise definition of our market price paths.

Definition 3.1** (Market price paths).**

Define the space of market price paths,

[TABLE]

where $\widehat{X}_{t}:=(t,X_{t})$ denotes the augmentation of $X$, $\widehat{\mathbb{X}}^{<\infty}$ is the signature of $\widehat{X}$ and the closure is taken under the $p$-variation distance, [LCL07, Definition 1.5]. The space of lead-lag market price paths is defined by

[TABLE]

where $\widehat{\mathbb{X}}^{LL,<\infty}$ denotes the signature of the lead-lag path associated to $(\widehat{\mathbb{X}}^{\leq 2},\langle\widehat{X}\rangle)$, as defined in Definition 2.14. Given $\widehat{\mathbb{X}}^{LL,<\infty}\in\widehat{\Omega}_{T}^{LL}$, we denote by $\widehat{\mathbb{X}}^{<\infty}\in\widehat{\Omega}_{T}$ the projection of $\widehat{\mathbb{X}}^{LL,<\infty}$ to $\widehat{\Omega}_{T}$.

So far, we have not imposed any probability measure on the market. We have only introduced the space of paths that will form the market, $\widehat{\Omega}_{T}^{LL}$. In Section 4 we will evaluate certain trading strategies by their performance in the market, for which we will use a probability measure. For this purpose, we will now define a probability space on the market paths. Most of the results in this paper, however, are not dependent on the probability measure.

Definition 3.2**.**

Consider the Borel $\sigma$-algebra $\mathcal{B}(\widehat{\Omega}_{T}^{LL})$, and define by $\mathbb{F}=\{\mathcal{F}_{t}\}_{t\in[0,T]}$ the filtration generated by the price path $X$. Let $\mathbb{P}$ be a probability measure on $(\widehat{\Omega}^{LL},\mathcal{B}(\widehat{\Omega}^{LL}))$ such that $\mathbb{E}[\widehat{\mathbb{X}}^{LL,\leq N}]$ is finite for all $N\in\mathbb{N}$. We will then consider the completed filtered probability space $(\widehat{\Omega}_{T}^{LL},\mathcal{B}(\widehat{\Omega}_{T}^{LL}),\mathbb{F},\mathbb{P})$.

We will not assume any particular model on the price – our approach is, in that sense, model-free. One could be interested, however, in imposing a particular model on the market, such as a certain semimartingale $X:[0,T]\to\mathbb{R}$. As discussed in Example 2.15, we can associate the semimartingale with the lead-lag path $(\widehat{\mathbb{X}}^{\leq 2},\langle\widehat{X}\rangle)$ where $\langle X\rangle$ is the quadratic variation of $X$ and $\widehat{\mathbb{X}}^{\leq 2}$ is the level-2 signature of $\widehat{X}$ (introduced in Example 2.9). Then, we would consider the probability space $(\widehat{\Omega}_{T}^{LL},\mathcal{B}(\widehat{\Omega}_{T}^{LL}),\mathbb{F},\mathbb{P})$ under which the coordinate process $X:[0,T]\to\mathbb{R}$ is a semimartingale with volatility given by the quadratic variation of $\widehat{X}$, i.e. $\langle\widehat{X}\rangle$.

Remark 3.3*.*

The assumption that $\mathbb{E}[\widehat{\mathbb{X}}_{0,T}^{LL,\leq N}]$ exists for all $N\in\mathbb{N}$ is very mild, and it is an infinite-dimensional version of the “moments of all order exist” statement for finite-dimensional random variables.

3.2 Payoff functions

Financial derivatives are given in terms of a payoff function that depends on the underlying asset(s). We will now make a precise definition of a payoff function.

Definition 3.4** (Payoff function).**

A payoff is defined as a Borel-measurable function $\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$. A payoff $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$ is said to be an $L^{q}$-payoff for $q\geq 1$ if $\mathbb{E}[|F|^{q}]<\infty$.

The definition above essentially defines a payoff function as any $\mathcal{F}_{T}$-measurable random variable. The financial interpretation is that, given a realisation of the price path, the holder of the derivative with payoff $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$ is paid $F(\widehat{\mathbb{X}}^{LL,<\infty})$ at time $T$.

Example 3.5*.*

Examples of payoff function include European options, American options, Asian options, lookback options, barrier options, futures, variance swaps, cliquet options, etc.

An important class of payoff functions, that will be used extensively in this paper, are linear signature payoff functions ([PA18]):

Definition 3.6** (Linear signature payoff).**

We say that a payoff $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$ is a linear signature payoff function is there exists a linear functional $f\in T((\mathbb{R}^{4})^{\ast})$ such that

[TABLE]

These signature payoffs will play a similar role to Arrow-Debreu primitive securities. As we will see, path-dependent exotic payoffs can be well-approximated by these linear signature payoffs. Notice that because the signature is defined as certain iterated integrals against the path, linear signature payoffs effectively contain in particular the P&L of all dynamic hedging strategies. Therefore, in a way, it is unsurprising that the class of linear signature payoffs is big and that they form a family of primitive securities.

Example 3.7*.*

We will now give a few examples of payoffs that can be written exactly as linear signature payoffs. Recall the word notation introduced in Section 2.1.

Let $K\in\mathbb{R}$, and set f=(1-K){\color[rgb]{0,0,1}\mathbf{\varnothing}}+{\color[rgb]{0,0,1}\mathbf{2}}. Then, $\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle=1-K+X_{T}-X_{0}=X_{T}-K$. In other words, the signature payoff is a forward with delivery price $K$. 2. 2.

Let $K\in\mathbb{R}$. Set f=(1-K){\color[rgb]{0,0,1}\mathbf{\varnothing}}+\frac{1}{T}{\color[rgb]{0,0,1}\mathbf{21}}. Then, $\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle=1-K+\frac{1}{T}\int_{0}^{T}(X_{s}-X_{0})ds=\frac{1}{T}\int_{0}^{T}X_{s}ds-K$. Therefore, Asian forwards are also signature payoffs.

3.3 Trading strategies

Intuitively, a trading strategy specifies the position that must be held by the trader at each time, given the observation of the price path up to that time. Moreover, this must be done in a non-anticipative way – in other words, traders are allowed to trade based on the past, but not the future. This idea is captured in the definition of trading strategies below.

Definition 3.8**.**

Define $\Lambda_{T}:=\bigcup_{t\in[0,T]}\widehat{\Omega}_{t}$, which is a metric space for a certain distance. The space of trading strategies is defined by $\mathcal{T}(\Lambda_{T}):=C(\Lambda_{T};\mathbb{R})$. We also denote by $\mathcal{T}^{q}(\Lambda_{T})$ the space of trading strategies with the following integrability condition:

[TABLE]

Remark 3.9*.*

The space $\Lambda_{T}$ is the space of signatures of all stopped paths. A similar space was discussed in [CF13, AC17, Gal94, BCC*+*16] and in [Dup09, BCH*+*17, Rig16] in the context of finance.

Intuitively, the space of trading strategies from Definition 3.8 essentially consists of all non-anticipative processes with respect to the filtration generated by $X$. Again, this emphasises the crucial condition in finance that one is only allowed to trade based on the past.

We will now define an important subspace of the space of trading strategies – namely, the space of linear signature trading strategies.

Definition 3.10** (Linear signature trading strategies).**

The space of linear signature trading strategies is given by

[TABLE]

It turns out that, in some sense, trading strategies can be approximated arbitrarily well by signature trading strategies. Therefore, if one is looking for an optimal trading strategy in $\mathcal{T}(\Lambda_{T})$, one could look for an optimal trading strategy in $\mathcal{T}_{sig}(\Lambda_{T})$ instead. This will be made more precise later on.

Given a trading strategy $\theta\in\mathcal{T}$, the profits and losses (P&L) associated to it is given by the rough path integral (see [LCL07]) given by $\int_{0}^{T}\theta(\widehat{\mathbb{X}}|_{[0,t]}^{<\infty})dX_{t}$. In the particular case of semimartingales, this integral agrees with the classical Itô integral – see [FHL16]. For instance, we have the following lemma, according to which Itô integrals of semimartingales of linear signature trading strategies are linear functions on the signature of the lead-lag path.

Lemma 3.11**.**

Let $X$ be a $d$-dimensional continuous semimartingale. Let $\ell\in T((\mathbb{R}^{2})^{\ast})$. Then, we have:

[TABLE]

where the integral is in the sense of Itô, the notation \ell{\color[rgb]{0,0,1}\mathbf{4}}\in T((\mathbb{R}^{4})^{\ast}) means the concatenation of the word associated to $\ell$ with the letter {\color[rgb]{0,0,1}\mathbf{4}} (introduced in Section 2.1) and $\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}$ is the signature of the (4-dimensional) lead-lag process, as defined in Definition 2.14.

Therefore, the profit of a trading strategy, defined with a rough path integral, agrees with the classical definition in terms of an Itô integral in the case of semimartingales (see [FHL16]) and in the particular case of linear signature trading strategies, the previous lemma states that the profits and losses – defined as an Itô integral against the semimartingale – is a linear functional of the signature of the lead-lag path.

4 Optimal hedging

In this section we study the following optimal polynomial hedging problem.

Definition 4.1** (Optimal polynomial hedging problem).**

Let $P\in\mathbb{R}[x]$ be a polynomial of degree $q\in\mathbb{N}$. Let $F$ be an $L^{q}$-payoff that pays at terminal time $T$ an amount of $F(\widehat{\mathbb{X}}^{LL,<\infty})$ with $\widehat{\mathbb{X}}^{LL,<\infty}\in\widehat{\Omega}_{T}^{LL}$. Let $p_{0}\in\mathbb{R}$ the initial capital. The associated optimal polynomial hedging problem (PHP) is to find a minimising sequence for:

[TABLE]

In the particular case where $P(x):=x^{2}$, (PHP) is of the form of the well-studied mean-variance optimal hedging problem ([Sch10, DR91, Sch92, DMKR95]). Writing the optimal control in terms of a polynomial $P$ will allow us to extend (PHP) to the exponential utility function as well.

Our objective will be to provide a numerical approach to finding a minimising sequence for the optimal hedging problem. We will tackle (PHP) by studying a linearised version of the problem. As we will see in Section 4.2, solving this sub-problem will be sufficient to solve the original hedging problem (PHP).

4.1 A signature linearisation of the problem

Problem (PHP) will be solved by solving the following optimisation sub-problem instead:

Definition 4.2** (Optimal linear signature hedging problem).**

Let $P\in\mathbb{R}[x]$ be a polynomial of degree $q\in\mathbb{N}$. Let $f\in T((\mathbb{R}^{4})^{\ast})$ and consider the associated linear $f$-signature payoff (Definition 3.6). Define the optimal linear signature hedging problem (LSHP) as finding a minimising sequence for

[TABLE]

As we will see in Section 4.2, being able to solve (LSHP) will be sufficient to solve (PHP). (LSHP), on the other hand, can be rewritten as a simpler optimisation problem that is numerically easier to solve:

Theorem 4.3**.**

Let $f\in T((\mathbb{R}^{4})^{\ast})$ and $p_{0}\in\mathbb{R}$. Let $P\in\mathbb{R}[x]$ be a polynomial of one variable. Then, the solution of the optimal linear signature hedging problem (LSHP) is given by the solution of the following polynomial optimisation problem:

[TABLE]

The optimisation problem (1) has two components. First, there is a linear functional, which depends the control $\ell\in T((\mathbb{R}^{2}))$ over which we are optimising, but does not depend on the price path $X$. The second component is the expected signature of the lead-lag process $\mathbb{E}\left[\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\right]$, which clearly depends on the price path $X$, but does not depend on the control. Notice that if a risk-neutral measure is used instead of the real-world probability measure, knowing the expected signature is equivalent of knowing the prices of all signature payoffs.

So far we haven’t made any assumptions on the price path $X$ nor the volatility $\langle X\rangle$. Hence, the only information one needs about the process to optimally hedge an exotic derivative is the expected signature of the lead-lag path corresponding to the underlying asset. In Section 6.1 we will see that one can infer the implied expected signature from market prices of exotic derivatives in a model-free way – therefore, no model has to be imposed on the price path. However, if one wants to assume a particular model for the price process (such as a particular Itô process or semimartingale), the expected signature can be computed either using Monte Carlo methods or, in certain cases, by solving a PDE. A more detailed discussions about this will be made in Section 4.3.

A consequence of Theorem 4.3 is the following corollary, which gives sufficient conditions for a linear signature payoff to be attainable or replicable.

Corollary 4.4**.**

Let $f\in T((\mathbb{R}^{4})^{\ast})$. Assume that $f$ is of the form

[TABLE]

where $N\in\mathbb{N}$ and p_{0},\lambda_{{\color[rgb]{0,0,1}\mathbf{w}}}\in\mathbb{R}. Then, $f$ is attainable and the optimal hedging strategy is given by the linear signature trading strategy

[TABLE]

4.2 The general optimal hedging problem

The optimisation problem (1) offers an implementable way of computing the optimal hedge in (LSHP) – for example, in the case of mean-variance hedging where $P(x):=x^{2}$, the optimisation problem (1) is reduced to finding the global minimum of a high-dimensional quadratic polynomial, which in turn can be easily found by solving a certain system of linear equation (see Section 4.3 for a discussion on numerically solving (LSHP) in practice).

However, the ultimate goal is to solve the optimal hedging problem shown in (PHP). In what follows, we justify why we may replace a general payoff function $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$ with a signature payoff, and why we may restrict the class of trading strategies from $\mathcal{T}^{q}(\Lambda_{T})$ to $\mathcal{T}_{sig}(\Lambda_{T})$. This section culminates in Theorem 4.7, which allows us to consider the tractable optimal hedging problem (LSHP) (and hence (1)) instead of the original nonlinear, hard-to-solve problem (PHP).

See Appendix C for the detailed justification of why solving (LSHP) is enough to solve (PHP).

From payoffs to signature payoffs

Arrow-Debreu securities are securities that pay 1 if a certain state of the market occurs, and nothing otherwise. Hence, exotic derivatives can be decomposed as linear combinations of such securities – Arrow-Debreu securities are, in other words, primitive securities from which all other securities are built.

In a similar fashion, the iterated integrals that define signatures (Definition 2.7) are primitive securities in the sense that other exotic, path-dependent payoffs are well-approximated by linear combinations of such iterated integrals. They are, effectively, basic securities from which other securities are built. Moreover, given that signature payoffs are defined as linear combinations of certain iterated integrals, they include a lot of information about the P&L of all possible dynamic trading strategies.

This is made precise in the following proposition ([PA18, Theorem 4.1]).

Proposition 4.5**.**

Let $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$ be a continuous payoff. Given any $\varepsilon>0$, there exists a compact set $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{T}$ (which does not depend on $F$) and $f\in T((\mathbb{R}^{4})^{\ast})$ such that:

$\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$, 2. 2.

$|F(\widehat{\mathbb{X}}^{LL,<\infty})-\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{LL,<\infty}\in\mathcal{K}_{\varepsilon}.$**

In other words, there exists a large compact set – large in the sense that with very high probability, all price paths one observes lie on that compact set – such that on the compact set, all continuous payoffs look like signature payoffs.

The authors want to emphasise that the linear functional $f$ from Proposition 4.5 does not depend on any model for the underlying assets. Indeed, it is a pathwise and a model-free density result that does not require any probability structure. The only role of the probability measure $\mathbb{P}$ in Proposition 4.5 is providing a notion of big compact sets – i.e. point 1. in Proposition 4.5.

From trading strategies to signature trading strategies

In Section 3.3, we defined the space of trading strategies $\mathcal{T}$ – which intuitively consists of all adapted processes – as well as the subspace of signature trading strategies $\mathcal{T}_{sig}\subset\mathcal{T}$. Similarly to signature payoffs, the space of signature trading strategies is big in the sense that arbitrary trading strategies can be well-approximated by them:

Proposition 4.6**.**

Let $\mathcal{K}\subset\Lambda_{T}$ be a compact set. Then, given any trading strategy $\theta\in\mathcal{T}$, there exists $\ell\in T((\mathbb{R}^{2})^{\ast})$ such that

[TABLE]

Putting everything together: from optimal hedging to optimal signature hedging

A consequence of Proposition 4.5 and Proposition 4.6 is the following theorem, which justifies why we can consider the optimal linear signature problem (LSHP) instead of the original optimal hedging problem (LSHP).

Theorem 4.7**.**

Let

[TABLE]

be the infimum of the optimal polynomial hedging problem (PHP). Given any $\varepsilon>0$, there exists a compact set $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{T}$, a linear signature payoff given by $f\in T((\mathbb{R}^{4})^{\ast})$ and a linear signature trading strategy given by $\ell\in T((\mathbb{R}^{2})^{\ast})$ such that:

$\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$, 2. 2.

$|F(\widehat{\mathbb{X}}^{LL,<\infty})-\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{LL,<\infty}\in\mathcal{K}_{\varepsilon}$, 3. 3.

$|\theta(\widehat{\mathbb{X}}|_{[0,t]}^{<\infty})-\langle\ell,\widehat{\mathbb{X}}_{0,t}^{<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{<\infty}\in\mathcal{K}_{\varepsilon}$* and $t\in[0,T]$,* 4. 4.

$|a_{\varepsilon}-a|\leq\varepsilon$, where

[TABLE]

That is, being able to solve the optimal linear signature hedging problem (LSHP) provides a numerically feasible way of finding a minimising sequence of the optimal polynomial hedging problem (PHP).

4.3 Solving the optimal linear signature hedging problem

Theorem 4.3, together with Theorem 4.7, offers an implementable approach to numerically approximating the optimal hedging strategy for a polynomial $P$ of degree $q$, an $L^{q}$-payoff $F$ and initial cash $p_{0}$.

The only information that is needed about the process is its expected signature. The expected signature plays a role similar to the moments of a real-valued random variable, but on path space – i.e. under certain growth assumptions, the expected signature of a process determines the law of the process [CL16]. Therefore, the fact that the optimal hedge depends on the expected signature essentially means that it depends on the whole law of the dynamics of the price path.

For obvious computational reasons, one cannot work with the whole expected signature – one has to begin by fixing a signature order $N\in\mathbb{N}$ and considering the corresponding truncated signature of the lead-lag process $\mathbb{E}[\widehat{\mathbb{X}}_{0,T}^{LL,\leq N}]$. We will show in Section 6 that, if one has access to market prices of enough exotics, it is possible to infer the market expected signature – i.e. the implied expected signature. In other words, it is possible to solve the optimal linear signature hedging problem (LSHP) in a model-free way, without imposing any model on the price dynamics of the underlying. If one wishes to impose a particular model on the market, however, the expected signature can be computed using Monte Carlo methods or even by solving a certain PDE – see [Ni12].

Remark 4.8*.*

The shuffle product of a word of length $n$ and a word of length $m$ is a sum of words of lengths $m+n$. Therefore, when considering the truncated signature $\mathbb{E}[\widehat{\mathbb{X}}_{0,T}^{LL,\leq N}]$ and a polynomial $P$ of degree $q$, (1) has to be minimised over $\ell\in T^{(\lfloor N/q\rfloor)}((\mathbb{R}^{2})^{\ast})$ rather than $\ell\in T((\mathbb{R}^{2})^{\ast})$.

Once (1) is restricted to a signature order $N\in\mathbb{N}$, the optimisation problem (1) is reduced to finding the minimum of a high-dimensional polynomial of degree $q$. In particular, in the mean-variance optimal hedging problem where the polynomial is given by $P(x):=x^{2}$, solving (1) consists of solving a system of linear equations.

Algorithm 1 describes the proposed algorithm. We make some practical remarks. Signatures can be computed using the publicly available software esig222https://pypi.org/project/esig/. An alternative package is iisignature333https://github.com/bottler/iisignature. Finally, the recursive definition of the shuffle product in Definition 2.4 allows for an easy and efficient implementation.

Once we have found a minimising linear functional $\ell\in T^{(\lfloor N/q\rfloor)}((\mathbb{R}^{2})^{\ast})$, the hedging strategy would be given by

[TABLE]

5 Extensions

In this section we will discuss a few extensions of our original framework. Needless to say, these extensions can be combined depending on the features one wishes the optimal hedging problem to have.

5.1 Exponential hedging

Often, one is interested in only hedging unfavourable differences between the hedged derivative and the hedging strategy. In other words, one wants to penalise losses and reward profits. This can be accomplished by considering the exponential hedging problem, where instead of minimising the expectation of a polynomial on the P&L, one replaces the polynomial by an exponential function $x\mapsto\exp(-\lambda x)$ for some risk parameter $\lambda>0$.

We will begin by introducing the space of trading strategies that we will consider admissible.

Definition 5.1** (Admissible trading strategy for exponential hedging).**

Define the space of admissible trading strategies

[TABLE]

We may now define the optimal exponential hedging problem as follows:

Definition 5.2** (Optimal exponential hedging problem).**

Let $\lambda>0$. Let $F$ be a payoff that is bounded a.s. Let $p_{0}\in\mathbb{R}$. The associated optimal exponential problem is:

[TABLE]

The parameter $\lambda>0$ measures the risk tolerance of the trader: the greater it is, the less tolerant the trader is with respect to losses.

The following proposition shows that we can tackle the optimal hedging problem (2) by reducing it to an optimal hedging problem of the form (LSHP), which was solved in Section (PHP).

Proposition 5.3**.**

Let

[TABLE]

be the infimum of the optimal exponential hedging problem. Given any $\varepsilon>0$, there exists a polynomial $P_{\varepsilon}\in\mathbb{R}[x]$, a compact set $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{T}$, a linear signature payoff given by $f\in T((\mathbb{R}^{4})^{\ast})$ and a linear signature trading strategy given by $\ell\in T((\mathbb{R}^{2})^{\ast})$ such that:

$P_{\varepsilon}\xrightarrow{\varepsilon\rightarrow 0}\exp(-\lambda\,\cdot)$* uniformly on compacts,* 2. 2.

$\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$, 3. 3.

$|F(\widehat{\mathbb{X}}^{LL,<\infty})-\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{LL,<\infty}\in\mathcal{K}_{\varepsilon}$, 4. 4.

$|\theta(\widehat{\mathbb{X}}|_{[0,t]}^{<\infty})-\langle\ell,\widehat{\mathbb{X}}_{0,t}^{<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{<\infty}\in\mathcal{K}_{\varepsilon}$* and $t\in[0,T]$,* 5. 5.

$|a_{\varepsilon}-a|\leq\varepsilon$, where

[TABLE]

5.2 Semi-static hedging

In certain situations, one wants to hedge a payoff $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$, for which one has access to a (finite) basket of derivatives $\mathcal{B}=\{G_{i}\}_{i=0}^{k}$ that are allowed for static hedging, as well as the underlying asset $X$ that can be used for dynamic hedging. In other words, at inception $t=0$ the trader has to form a portfolio on the basket $\mathcal{B}$ and then the trader can dynamically trade on the underlying $X$. For example, the payoff $F$ we wish to hedge could be an exotic option, and the basket $\mathcal{B}$ could be a basket of simpler vanilla options. We will assume that the payoff $F$, as well as each payoff $G_{i}\in\mathcal{B}$, are $L^{q}$ payoffs. Moreover, following Section 4.1 and Section 4.2, we will assume that $F$ is a $f$-signature payoff and $G_{i}$ is a $g_{i}$-signature payoff, with $f,g_{i}\in T((\mathbb{R}^{4})^{\ast})$.

If we allow semi-static hedging, the optimal hedging problem is then defined as:

[TABLE]

where $\Gamma\subset\mathbb{R}^{k}$ determines the region of admissible strategies on $\mathcal{B}$. For example, if no constraints are imposed, one could choose $\Gamma=\mathbb{R}^{k}$. If no short-selling is allowed, on the other hand, one would choose $\Gamma=\mathbb{R}_{+}^{k}$. Other choices are also allowed, which can include liquidity constraints or other more complex, inter-connected constraints on $\mathcal{B}$.

Then, the semi-static optimal hedging problem is then reduced to the following.

Corollary 5.4**.**

Given a basket of signature payoffs $\mathcal{B}=\{g_{i}\}_{i=1}^{k}$ and a region $\Gamma\subset\mathbb{R}^{k}$, the solution of the semi-static optimal hedging problem is given by the solution of

[TABLE]

5.3 Adding transaction costs

It will turn out, unsurprisingly, that linear signature trading strategies with wild oscillations will incur infinite transaction costs. This could be avoided by considering linear signature trading strategies that have a well-defined speed of trading:

[TABLE]

The function $\langle v,\widehat{\mathbb{X}}_{0,t}^{<\infty}\rangle$ indicates the trading speed – i.e. the amount of the underlying asset that will be bought or sold at each time $t$. For such a choice of trading speed, the trader’s position on the asset at time $t$ will be $\int_{0}^{t}\langle v,\widehat{\mathbb{X}}_{0,u}^{<\infty}\rangle du$. Therefore, such trading strategies will be differentiable and hence they will not incur infinite transaction costs.

We will now introduce the following fixed and proportional quadratic costs for trading strategies in $\mathcal{T}_{speed}$.

Definition 5.5** (Fixed quadratic transaction costs).**

Consider a speed of trading $v\in T((\mathbb{R}^{2})^{\ast})$. We define the fixed quadratic costs with intensity $\alpha\geq 0$ incurred by $v$ along $\widehat{\mathbb{X}}^{LL,<\infty}\in\widehat{\Omega}_{T}$ as

[TABLE]

Definition 5.6** (Proportional quadratic transaction costs).**

Consider a speed of trading $v\in T((\mathbb{R}^{2})^{\ast})$. The proportional quadratic costs with intensity $\alpha\geq 0$ incurred by $v$ along $\widehat{\mathbb{X}}^{LL,<\infty}\in\widehat{\Omega}_{T}$ is then defined as

[TABLE]

Then, one can naturally modify (LSHP) to include fixed quadratic costs,

[TABLE]

or proportional transaction costs,

[TABLE]

We then have the following corollaries of Theorem 4.3.

Corollary 5.7**.**

The solution of the optimal hedging problem under fixed quadratic trading costs (3) is given by the solution of the following optimisation problem:

[TABLE]

Similarly, the solution of the optimal hedging under proportional transaction costs (4) is given by

[TABLE]

5.4 Liquidity constraints

So far, we have implicitly assumed that the market for the underlying asset is perfectly liquid: we can, at any given time, take a long or short position of any size. However, this assumption may not be realistic for some assets, so that one has to impose certain liquidity constraints. We will do so by imposing a certain boundedness condition on the speed of trading that was introduced in Section 5.3. More specifically, we will consider all trading speeds $v\in T((\mathbb{R}^{2})^{\ast})$ such that $\lVert v\rVert\leq M$, for some illiquidity constant $M\geq 0$. This parameter could be estimated from historical data of the asset, for example.

Under liquidity constraints, the unconstrained optimal hedging problem (LSHP) is transformed to the following:

Definition 5.8** (Optimal hedging problem with liquidity constraints).**

Given an illiquidity constant $M\geq 0$, the following problem is defined as the optimal hedging strategy with liquidity constraint $M$:

[TABLE]

The solution of the optimal hedging problem with liquidity constraints is then a constrained optimisation problem:

Corollary 5.9**.**

Let $M\geq 0$ be an illiquidity constant. The solution of the optimal hedging problem with liquidity constraint $M$ is given by

[TABLE]

5.5 Delayed hedging

Suppose a trader wishes to hedge a certain derivative given by the payoff $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$, whose lifespan is $[0,T]$. However, the trader is at time $t>0$, so that she cannot follow the hedging strategy provided by Theorem 4.3. What is the optimal strategy that the trader can carry?

Again, by Section 4.2 we will assume that $F$ is a $f$-signature payoff for $f\in T((\mathbb{R}^{4})^{\ast})$. The objective is then to solve the following optimal hedging problem:

[TABLE]

where $p_{t}$ is $\mathcal{F}_{t}$-measurable and represents the cash held at time $t$. Notice that

[TABLE]

We then have:

Corollary 5.10**.**

The solution of the optimal hedging problem (7) where the trader starts hedging at time $t>0$ is given by

[TABLE]

In other words, all the trader has to do is compute the signature up to time $t$ of the lead-lag process of the augmented price path, as well as the expected signature for the remaining interval $[t,T]$, and then solve the optimisation problem (8).

6 Pricing and hedging from market data: numerical experiment

In Section 4 we showed that to solve the optimal hedging (PHP) it is sufficient to consider the linearised version of the problem (LSHP). In Theorem 4.3, the problem was then reduced to (1).

To solve (1), the only information that is needed about the underlying process is its expected signature. Therefore, an interesting question one could ask is whether it is possible to somehow estimate the expected signature from the market. In this section, we will study how we could use market data to estimate the expected signature of the market. More specifically, we will show that we can infer the expected signature that matches market prices of exotic payoffs – namely, the implied expected signature. Because we will be using prices of derivatives, we will be working under a risk-neutral measure instead of the objective probability measure. The implied expected signature will then be used in Section 6.2 and Section 6.3 to price and hedge payoffs, respectively.

6.1 The implied expected signature

The volatility of the underlying asset is a relevant quantity when describing risk-neutral measures. In some cases, volatility is all one needs to know to price certain options.

However, volatility on its own only captures some aspects of risk-neutral measures on path space, and it does not characterise them. One needs a much richer object if one attempts to determine risk-neutral measures. It turns out that, under certain conditions, the expected signature fully characterises probability measures on path space ([CL16]). Therefore, one only needs to know the expected signature of a measure to completely describe it.

Let $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$ be a payoff whose price is observable in the market. Following [PA18] and by Proposition 4.5, we approximate the price of a payoff $F:\widehat{\Omega}_{T}^{LL}\rightarrow\mathbb{R}$, which is given by $Z_{T}\mathbb{E}^{\mathbb{Q}}[F(\widehat{\mathbb{X}}^{LL,<\infty})]$ with $Z_{T}$ the discount factor for the interval $[0,T]$ and $\mathbb{Q}$ a risk-neutral measure, by a linear signature payoff (Definition 3.6):

[TABLE]

Implied volatility is defined as the volatility of the underlying asset that makes the model prices of certain vanilla options match the prices observed in the market. Similarly, we may define the implied expected signature as the expected signature that matches observed prices of exotic derivatives. The implied expected signature not only captures the implied volatility, but it also captures other aspects of the risk-neutral measure. Notice, moreover, that knowing the implied expected signature is equivalent to knowing the prices of all linear signature payoffs.

If one has access to market prices of a sufficiently varied range of payoffs, one can leverage this information to infer the implied expected signature. Indeed, assume that one has access to a family of pairs $\{(F_{i},p_{i})\}_{i}$ of payoffs $F_{i}$ with market prices $p_{i}$. We may replace each payoff $F_{i}$ by an approximating linear (truncated) signature payoff of order $N\in\mathbb{N}$ given by the functional $\ell_{i}\in T^{N}((\mathbb{R}^{4})^{\ast})$. Then, one has that

[TABLE]

One may then apply linear regression to estimate the (discounted) implied expected signature $Z_{T}\mathbb{E}^{\mathbb{Q}}\left[\widehat{\mathbb{X}}^{LL,\leq N}_{0,T}\right]$.

6.2 Pricing with the implied expected signature

In order to apply the procedure described in the last subsection, one would need to be able to observe market prices of a rich-enough class of payoffs. Although certain vanilla options are exchange-traded, most exotic derivatives are not. Therefore, obtaining market prices of sufficient exotic derivatives to induce the implied expected signature may be a challenge.

However, there are multiple data providers that offer consensus market prices of a range of OTC (Over-The-Counter) exotic derivatives. These prices reflect the consensus prices from market participants, and they can be seen as market prices for these exotic derivatives. The authors have followed the procedure from Section 6.1 on these consensus market prices, but it was made clear by the data provider that “(publishing the results) is not a permitted use case for the (…) data”. Therefore, to illustrate the feasibility of the methodology presented in this section, we performed a numerical experiment where we assumed that we are exogeneously given (synthetic) prices of a number of derivatives. It is noteworthy to mention that results on the consensus market prices were similar to the results obtained on synthetic data.

In this experiment these prices come from a Heston model, which is completely unknown to the trader. In other words, although we have produced the payoff prices from a specific model, the agent or trader is completely ignorant of it and the only information she can leverage is the knowledge about the prices of a range of derivatives. This resembles the real-life situation where the trader is able to observe market prices for a variety of derivatives, but is ignorant of the market dynamics.

From the Heston model we simulated prices for 150 payoffs with maturity 1 year using an interest rate of $2\%$. The payoff types we considered were European options, barrier options and variance swaps (50 payoffs of each type were considered). We divided the set of 150 derivatives into a training set of 75 derivatives and a testing set of 75 derivatives. The size of the dataset, as well as the payoff types, were selected to make the dataset similar to the one offered by market consensus providers.

We used the training set to infer the discounted implied expected signature, following the procedure proposed in Section 6.1. The order of the truncated signature that was considered was $N=5$. We then used the computed discounted implied expected signature to predict the market prices of the derivatives in the testing set. These predicted prices were then compared to the real market prices. The results are shown in Figure 1. The predictions of the market prices seem to be quite accurate, with an $R^{2}$ of 0.99994.

Notice that, for any risk-neutral measure $\mathbb{Q}$, we have \left\langle{\color[rgb]{0,0,1}\mathbf{\varnothing}},Z_{T}\mathbb{E}^{\mathbb{Q}}\left[\widehat{\mathbb{X}}^{LL,\leq N}_{0,T}\right]\right\rangle=Z_{T}. Therefore, we can estimate the discount factor $Z_{T}$ from the discounted implied signature. From our dataset, we obtained the estimate $Z_{T}\approx 0.9802966$. This leads an estimation of the short rate of $-\log Z_{T}\approx 1.9900\%$, very close to the real short rate of $2\%$ that was used.

6.3 Hedging with the implied expected signature

Once we obtained the implied expected signature and validating its accuracy at obtaining market prices for exotic payoffs, we proceeded to apply Algorithm 1 with this implied expected signature on different payoffs. In all cases, we considered the mean-variance hedging problem by taking the polynomial $P(x):=x^{2}$ in Algorithm 1, and the initial capital $p_{0}$ was set to market price of each derivative. The signature order was set to 5, as in Section 6.2. Daily rebalancing was used.

Given that the Heston model is incomplete and we are only allowing daily rebalancing, we know that perfect hedging is not possible in general. Figure 7 shows the P&L of the hedged portfolio corresponding to various payoffs.

7 Experiments on synthetic data

Our methodology is intrinsically model-free, in the sense that we do not assume any particular model for the market dynamics, other than the price path follows a continuous semimartingale. The only information that is needed is the expected signature, which as shown in Section 6 it can be estimated from market prices of exotic derivatives. However, in certain settings one does want to impose a model on the price path. For example, a bank may want to learn how to hedge an exotic payoff when the market dynamics are given by one of the internal models of the bank. As it was discussed in Section 4.3, if a particular model is used for the market dynamics one is then able to estimate the expected signature, and the methodology proposed in this paper can therefore be applied.

In this section we implement the proposed approach in a wide range of examples to show the effectiveness of the methodology on different market models. We will begin by considering in Section 7.1 a toy example with a simple payoff in a complete market, in order to compare the signature hedging strategy with the (known) replicating strategy. Then, in Section 7.2 we will implement our methodology on path-dependent payoffs in an incomplete market. In Section 7.3 we will consider the exponential hedging problem and finally in Section 7.4 we will study the hedging problem under transaction costs.

7.1 Toy example

First, we considered the simple case where we assume that $X$ follows a Black–Scholes model and we want to hedge the derivative with payoff $F(\widehat{\mathbb{X}}^{LL,<\infty})=X_{T}^{2}$ at terminal time $T$. This payoff, under this model, is attainable and we should therefore be able to perfectly hedge it. Given that we can explicitly find what the replicating strategy should be – it is the delta hedge – this example will be useful to determine whether the optimal signature strategy matches the replicating strategy.

We implemented Algorithm 1 for the polynomial $P(x):=x^{2}$, so that we are considering the mean-variance hedging problem. The initial capital $p_{0}$ was taken to be the risk-neutral price for the payoff, and the signature order was set to 8. We fixed the maturity to 1 year $(T=1)$, and we assumed daily rebalancing.

Figure 8 shows the optimal hedging strategy (i.e. the replicating strategy) and the signature hedging strategy provided by Algorithm 1. As we see, both strategies match very well. When we consider the P&L of the replicating and signature strategies (see Figure 9) we observe that they have a very similar performance.

7.2 Path-dependent payoffs on the Heston model

We now consider path-dependent payoffs on the Heston model, which is incomplete. The payoffs we took were Asian options, barrier call options, lookback options and variance swaps. As in the previous section, we considered the mean-variance hedging problem with maturity 1 year, and we set $p_{0}$ to be the risk-neutral price for each payoff. Again, the signature order we considered was 8.

Figure 15 shows the P&L of the hedged portfolio at maturity with daily rebalancing. Ideally, the payoffs would be perfectly hedged so that the P&L of the hedged portfolio would be identically zero. However, given that the Heston model is incomplete and we are considering daily rebalancing, this is not possible in general.

7.3 Exponential hedging

If we change the risk preferences of the trader in order to penalise losses but not profits, we may consider the exponential hedging problem rather than the mean-variance hedging problem. Following Section 5.1, we approximate $x\mapsto\exp(-\lambda x)$ by polynomials. We then solve the optimal linear signature hedging problem for the Asian option payoff. Notice that this payoff is not a.s. bounded and it therefore does not satisfy the hypotheses of Proposition 5.3. However, we can overcome this issue by assuming that the payoff was truncated on $[-M,M]$, for $M>0$ large enough.

The performance of the signature hedging strategy is shown in Figure 16, where the risk parameter $\lambda=0.25$ was considered. Notice that this risk parameter has shifted the P&L profile from Figure 11, reflecting the change in the trader’s risk preferences.

7.4 Transaction costs

To study the effect of transaction costs, we consider the payoff $F(\widehat{\mathbb{X}}^{LL,<\infty})=X_{T}^{2}$ that was studied in Section 7.1. We added fixed quadratic transaction costs (Definition 5.5) with $\alpha=10^{-6}$ and we compared the performance of the signature hedging strategy, obtained by solving (5).

Figure 17 shows that the P&L of the replicating hedging strategy drops drastically when transaction costs are added, whereas the P&L of the signature hedging strategy is much less affected by these transaction costs.

8 Conclusion

In this paper we introduce a family of primitive securities called signature payoffs (Definition 3.6). In the spirit of Arrow-Debreu, these payoffs approximate arbitrarily well other exotic, path-dependent derivatives. Because signature payoffs are defined as linear combinations of certain iterated integrals, the family of all signature derivatives includes a lot of information about the P&L of all possible dynamic trading strategies.

In Section 4, we show that these signature payoffs can be used to reduce the original hard-to-solve optimal hedging problem (PHP) to a polynomial optimisation problem that is numerically easy to solve, (1). The only information about the underlying process that is needed to accomplish this is its expected signature – which, in the case where a risk-neutral measure is used, is equivalent to knowing the prices of all signature payoffs. Moreover, our approach is intrinsically model-free – we do not need to impose any particular model on the market dynamics.

We also demonstrated that our methodology can be used in practice by pricing and hedging certain payoffs from market data, using the implied expected signatures (Section 6). We also explore in Section 7 the optimal hedging strategies produced by our methodology for different payoff functions when a particular market model is used.

Disclosure statement

Opinions and estimates constitute our judgement as of the date of this Material, are for informational purposes only and are subject to change without notice. This Material is not the product of J.P. Morgans Research Department and therefore, has not been prepared in accordance with legal requirements to promote the independence of research, including but not limited to, the prohibition on the dealing ahead of the dissemination of investment research. This Material is not intended as research, a recommendation, advice, offer or solicitation for the purchase or sale of any financial product or service, or to be used in any way for evaluating the merits of participating in any transaction. It is not a research report and is not intended as such. Past performance is not indicative of future results. Please consult your own advisors regarding legal, tax, accounting or any other aspects including suitability implications for your particular circumstances. J.P. Morgan disclaims any responsibility or liability whatsoever for the quality, accuracy or completeness of the information herein, and for any reliance on, or use of this material in any way.

Important disclosures at: www.jpmorgan.com/disclosures.

Appendix A The lead-lag path: practical considerations

In this appendix, we will discuss some practical considerations about how to compute the lead-lag path for discrete data, as well as for semimartingales.

Let $D=\{t_{i}\}_{i=0}^{n}\subset[0,T]$ be a finite partition, and let $Z:D\to\mathbb{R}^{d}$ be discrete path. The lead-lag transformation of $Z$ is defined below.

Definition A.1**.**

[Lead-lag transformation, [FHL16, Definition 2.1]] The lead-lag transformation of $Z$ associated with $D$ is the $2d$-dimensional piecewise linear path $Z^{D,LL}:=(Z^{D,b},Z^{D,f}):[0,T]\to\mathbb{R}^{2d}$ defined by

[TABLE]

The component $Z^{D,b}$ is the lag or backward component, and $Z^{D,f}$ is the lead or forward component. By taking the signature of this piecewise linear path, we obtain the signature of the lead-lag path $\mathbb{Z}^{D,LL,<\infty}$.

The lead-lag transformation differentiates the role played by the past and the future. This is done by keeping track of the immediate past (the lag component) and the immediate future (the lead component).

In order to give an intuition of what the lead-lag transformation is, Figure 18 shows the lead-lag transformation of a certain price path. As the name suggests, the lead component is leading the lag component.

Now, let $Z:[0,T]\to\mathbb{R}^{d}$ be a continuous semimartingale with quadratic variation $\langle Z\rangle$. As discussed in Example 2.15, this induces a lead-lag path $(\mathbb{Z}^{\leq 2},\langle Z\rangle)$, whose signature is $\mathbb{Z}^{LL,<\infty}$. The lemma below provides a method to compute the signature of the lead-lag path of a semimartingale in practice: one can sample the semimartingale, compute the lead-lag transformation of the corresponding discrete path and then find its signature.

Lemma A.2**.**

[[FHL16, Theorem 4.1]] Let $Z:[0,T]\to\mathbb{R}^{d}$ be a continuous semimartingale. For each finite partition $D\subset[0,T]$, denote by $Z^{D}$ the corresponding lead-lag transformation and by $\mathbb{Z}^{D,LL,<\infty}$ its signature (Definition A.1). Let $\mathbb{Z}^{LL,<\infty}$ be the signature of the lead-lag path $(\mathbb{Z}^{\leq 2},\langle Z\rangle)$ associated with the semimartingale (Definition 2.14). Then,

[TABLE]

where the limit is taken in under the $p$-variation distance (see [FHL16]).

Appendix B Proofs

Lemma B.1** (Signature of a perturbed rough path).**

Let $\mathbb{X}\in G\Omega_{p}([0,T],\mathbb{R}^{d})$ be a $p$-rough path with $p\in[2,3)$, and let $\varphi:C([0,T];\mathfrak{so}(d))$ be of bounded variation. Define the second-level perturbation $\mathbb{Y}:=\mathbb{X}+\varphi\in G\Omega_{p}([0,T];\mathbb{R}^{d})$. Define $z^{1}:=X,z^{2}:=\varphi$. Given {\color[rgb]{0,0,1}\mathbf{I}}={\color[rgb]{0,0,1}\mathbf{i_{1}\ldots i_{k}}}\in\{{\color[rgb]{0,0,1}\mathbf{1}},{\color[rgb]{0,0,1}\mathbf{2}}\}^{k}, let

[TABLE]

Then, for each $N\geq 1$ the level $N$ signature of $\mathbb{Y}$ is given by

[TABLE]

Proof.

If $N=1$, the sum above is reduced to $a^{(1)}=\int_{0}^{T}dX_{0,t}=X_{0,T}=Y_{0,T}$, so that the claim holds. Assume that the statement is true for $N-1$. We will show that it also holds for $N\geq 2$.

By [Lyo14, Lemma 4.6], $\mathbb{Y}^{\leq N}$ satisfies the rough differential equation

[TABLE]

By [FV10, Theorem 12.16], the solution of the above rough differential equation is also the solution of the rough differential equation with drift

[TABLE]

where $[e_{i},e_{j}]:=e_{i}\otimes e_{j}-e_{j}\otimes e_{i}$ denotes the Lie bracket. Hence, the level $N$ projection of $\mathbb{Y}^{\leq N}$ will satisfy

[TABLE]

because $\varphi$ is antisymmetric. By induction hypothesis,

[TABLE]

Hence,

[TABLE]

as desired.

∎

Lemma 3.11.

Let $X$ be a $d$-dimensional continuous semimartingale. Let $\ell\in T((\mathbb{R}^{2})^{\ast})$. Then, we have:

[TABLE]

where the integral is in the sense of Itô, the notation \ell{\color[rgb]{0,0,1}\mathbf{4}}\in T((\mathbb{R}^{4})^{\ast}) means the concatenation of the word associated to $\ell$ with the letter {\color[rgb]{0,0,1}\mathbf{4}} (introduced in Section 2.1) and $\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}$ is the signature of the (4-dimensional) lead-lag process, as defined in Definition 2.14.

Proof.

For a $d$-dimensional path $Z$, define the $2d$-dimensional path $\overline{Z}:=(Z,Z)$, with the corresponding signature $\overline{\mathbb{Z}}^{<\infty}$. It was shown in [FHL16] that $\widehat{\mathbb{X}}^{LL,<\infty}$ is the signature of the perturbed rough path $\widehat{\mathbb{X}}^{LL,\leq 2}:=\overline{\widehat{\mathbb{X}}}^{\leq 2}+\psi$, with

[TABLE]

Let $N\geq 1$. We have

[TABLE]

Therefore, we have to show that

[TABLE]

We proceed by induction. If $N=1$, we have

[TABLE]

Since \langle\ell{\color[rgb]{0,0,1}\mathbf{4}},\psi\rangle=-\dfrac{1}{2}\left\langle\ell{\color[rgb]{0,0,1}\mathbf{4}},\langle\overline{\widehat{X}}\rangle\right\rangle, it follows that (9) holds for $N=1$.

Assume that (9) holds for $N$, we will show that it also holds true for $N+1$. By induction hypothesis, (9) is reduced to:

[TABLE]

By Lemma B.1,

[TABLE]

Notice that \langle\ell{\color[rgb]{0,0,1}\mathbf{4}},a^{\color[rgb]{0,0,1}\mathbf{I}}\rangle is nonzero only for {\color[rgb]{0,0,1}\mathbf{I_{1}}}=\underbrace{{\color[rgb]{0,0,1}\mathbf{11\ldots 1}}}_{N+2} and {\color[rgb]{0,0,1}\mathbf{I_{2}}}=\underbrace{{\color[rgb]{0,0,1}\mathbf{11\ldots 1}}}_{N}{\color[rgb]{0,0,1}\mathbf{2}}. Hence,

[TABLE]

as desired.

∎

Theorem 4.3.

Let $f\in T((\mathbb{R}^{4})^{\ast})$ and $p_{0}\in\mathbb{R}$. Let $P\in\mathbb{R}[x]$ be a polynomial of one variable. Then, the solution of the optimal linear signature hedging problem (LSHP) is given by the solution of the following polynomial optimisation problem:

[TABLE]

Proof.

By Lemma 3.11, (LSHP) will be given by:

[TABLE]

where $(\star)$ follows by the shuffle product property (Lemma 2.11).

∎

Proposition 5.3.

Let

[TABLE]

be the infimum of the optimal exponential hedging problem. Given any $\varepsilon>0$, there exists a polynomial $P_{\varepsilon}\in\mathbb{R}[x]$, a compact set $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{T}$, a linear signature payoff given by $f\in T((\mathbb{R}^{4})^{\ast})$ and a linear signature trading strategy given by $\ell\in T((\mathbb{R}^{2})^{\ast})$ such that:

$P_{\varepsilon}\xrightarrow{\varepsilon\rightarrow 0}\exp(-\lambda\,\cdot)$* uniformly on compacts,* 2. 2.

$\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$, 3. 3.

$|F(\widehat{\mathbb{X}}^{LL,<\infty})-\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}\in\mathcal{K}_{\varepsilon}$, 4. 4.

$|\theta(\widehat{\mathbb{X}}|_{[0,t]}^{<\infty})-\langle\ell,\widehat{\mathbb{X}}_{0,t}^{<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{<\infty}\in\mathcal{K}_{\varepsilon}$* and $t\in[0,T]$,* 5. 5.

$|a_{\varepsilon}-a|\leq\varepsilon$, where

[TABLE]

Proof.

Let $I\subset\mathbb{R}$ be a compact interval such that $p_{0}+\int_{0}^{T}\theta(\widehat{\mathbb{X}}|_{[0,t]}^{<\infty})dX_{t}-F(\widehat{\mathbb{X}}^{LL,<\infty})\in I$ a.s. Let $P_{\varepsilon}$ be the Taylor expansion of $\exp(-\lambda\;\cdot)$ around the origin of degree large enough so that $\lVert P_{\varepsilon}-\exp(-\lambda\;\cdot)\rVert_{L^{\infty}(I)}<\varepsilon$.

Let $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{p}$ compact be such that $\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$ and

[TABLE]

Take $\ell\in T((\mathbb{R}^{2})^{\ast})$, $f\in T((\mathbb{R}^{4})^{\ast})$ such that 3. and 4. hold, which we can do due to Proposition 4.5 and Proposition 4.6. Then, we have:

[TABLE]

By Propositions 4.5 and 4.6, $(\star)<\varepsilon$. Moreover, because $\lVert P_{\varepsilon}-\exp(-\lambda\;\cdot)\rVert_{L^{\infty}(I)}<\varepsilon$, we have $(\star\star)<\varepsilon$, and the proof follows. ∎

Corollary 5.7.

The solution of the optimal hedging problem under fixed quadratic trading costs (3) is given by the solution of the following optimisation problem:

[TABLE]

Similarly, the solution of the optimal hedging under proportional transaction costs (4) is given by

[TABLE]

Proof.

(11) follows from Theorem 4.3 and from the fact that

[TABLE]

Similarly, (12) follows from Theorem 4.3 and the fact that

[TABLE]

∎

Appendix C

In Section 4.2, we justified why solving the linear signature hedging problem (LSHP) allows us to numerically estimate the solution of the polynomial hedging problem (PHP).

The objective of this section is to give the technical details of why such an approximation is justified.

Proposition 4.5.

Let $F:\widehat{\Omega}_{T}^{LL}\to\mathbb{R}$ be a continuous payoff. Given any $\varepsilon>0$, there exists a compact set $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{T}$ (which does not depend on $F$) and $f\in T((\mathbb{R}^{4})^{\ast})$ such that:

$\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$, 2. 2.

$|F(\widehat{\mathbb{X}}^{LL,<\infty})-\langle f,\widehat{\mathbb{X}}_{0,T}^{<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{LL,<\infty}\in\mathcal{K}_{\varepsilon}.$**

Proof.

Let $f,g\in T((\mathbb{R}^{4})^{\ast})$. Then, by the shuffle product, we have:

[TABLE]

Therefore, linear signature payoffs form an algebra. Moreover, the uniqueness of the signature (Corollary 2.13) implies that the family of linear signature payoffs separate points. Also, they trivially contain constants.

Let $\varepsilon>0$. Given that $\widehat{\Omega}_{T}$ is separable, there exists $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{T}$ compact such that $\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$. Because the family of linear signature payoffs forms an algebra, separates points and contains constants, by Stone–Weierstrass theorem there exists a $\ell$-linear signature payoff, with $f\in T((\mathbb{R}^{4})^{\ast})$, such that $|F(\widehat{\mathbb{X}}^{LL,<\infty})-\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{LL,<\infty}\in\mathcal{K}_{\varepsilon}.$ ∎

Proposition C.1**.**

Set $1\leq q<\infty$, and let $F:\widehat{\Omega}_{T}^{LL}\rightarrow\mathbb{R}$ be an $L^{q}$-payoff. Given any $\varepsilon>0$, there exists a compact set $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{p}$ and a $f$-linear signature payoff with $f\in T((\mathbb{R}^{4})^{\ast})$ such that:

$\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$, 2. 2.

$\mathbb{E}[|F(\widehat{\mathbb{X}}^{LL,<\infty})-\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle|^{q}\;;\;\mathcal{K}_{\varepsilon}]<\varepsilon.$**

Proof.

Let $\varepsilon>0$. Since continuous functions are dense in $L^{q}$, there exists a continuous payoff $G:\widehat{\Omega}_{T}\to\mathbb{R}$ such that $\lVert F-G\rVert_{L^{q}}^{q}<\varepsilon/2$. By Proposition C.1 we may pick $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}$ and $f\in T((\mathbb{R}^{4})^{\ast})$ such that $\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$, $\mathbb{E}[|G(\widehat{\mathbb{X}}^{LL,<\infty})|^{q};\mathcal{K}_{\varepsilon}^{c}]<\varepsilon$ and

[TABLE]

Then,

[TABLE]

∎

Proposition 4.6.

Let $\mathcal{K}\subset\Lambda_{T}$ be a compact set. Then, given any trading strategy $\theta\in\mathcal{T}$, there exists $\ell\in T((\mathbb{R}^{2})^{\ast})$ such that

[TABLE]

Proof.

Given $\theta_{1},\theta_{2}\in\mathcal{T}_{\mathrm{sig}}(\mathcal{K})$, there exist $\ell_{1},\ell_{2}\in T((\mathbb{R}^{2})^{\ast})$ such that $\theta_{i}(\widehat{\mathbb{X}}|_{[0,t]})=\langle\ell_{i},\widehat{\mathbb{X}}_{0,t}^{<\infty}\rangle$ for each $\widehat{\mathbb{X}}|_{[0,t]}\in\mathcal{K}$ and $i=1,2$. Define $\theta_{1,2}(\mathbb{X}|_{[0,t]}):=\langle\ell_{1}\sqcup\mathchoice{\mkern-7.0mu}{\mkern-7.0mu}{\mkern-3.2mu}{\mkern-3.8mu}\sqcup\ell_{2},\mathbb{X}_{0,t}^{<\infty}\rangle$. Then,

[TABLE]

Hence, $\mathcal{T}_{\mathrm{sig}}(\mathcal{K})$ is an algebra. Given that it also separates points ([BGLY16]) and contains constants, it follows by Stone–Weierstrass theorem that given any trading strategy $\theta\in\mathcal{T}$, there exists $\ell\in T((\mathbb{R}^{2})^{\ast})$ such that

[TABLE]

∎

Theorem C.2**.**

Let

[TABLE]

be the infimum of the optimal polynomial hedging problem (PHP). Given any $\varepsilon>0$, there exists a compact set $\mathcal{K}_{\varepsilon}\subset\widehat{\Omega}_{T}$, a linear signature payoff given by $f\in T((\mathbb{R}^{4})^{\ast})$ and a linear signature trading strategy given by $\ell\in T((\mathbb{R}^{2})^{\ast})$ such that:

$\mathbb{P}[\mathcal{K}_{\varepsilon}]>1-\varepsilon$, 2. 2.

$|F(\widehat{\mathbb{X}}^{LL,<\infty})-\langle f,\widehat{\mathbb{X}}_{0,T}^{LL,<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{LL,<\infty}\in\mathcal{K}_{\varepsilon}$, 3. 3.

$|\theta(\widehat{\mathbb{X}}|_{[0,t]}^{<\infty})-\langle\ell,\widehat{\mathbb{X}}_{0,t}^{<\infty}\rangle|<\varepsilon\quad\forall\,\widehat{\mathbb{X}}^{<\infty}\in\mathcal{K}_{\varepsilon}$* and $t\in[0,T]$,* 4. 4.

$|a_{\varepsilon}-a|\leq\varepsilon$, where

[TABLE]

Proof.

Follows from Proposition C.1 and Proposition 4.6, and the triangle inequality. ∎